{
  "id": "2411.10767",
  "version": "v1",
  "published": "2024-11-16T10:17:42.000Z",
  "updated": "2024-11-16T10:17:42.000Z",
  "title": "Green's formula and Derived Hall algebras",
  "authors": [
    "Ji Lin"
  ],
  "comment": "24pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "The aim of this note is to clarify the relationship between Green's formula and the associativity of multiplication for derived Hall algebra in the sense of To\\\"{e}n (Duke Math J 135(3):587-615, 2006), Xiao and Xu (Duke Math J 143(2):357-373, 2008) and Xu and Chen (Algebr Represent Theory 16(3):673-687, 2013). Let $\\mathcal{A}$ be a finitary hereditary abelian category. It is known that the associativity of derived Hall algebra $\\mathcal{D}\\mathcal{H}_t(\\mathcal{A})$ implies Green's formula. We show the converse statement holds. Namely, Green's formula implies the associativity of the derived Hall algebra $\\mathcal{D}\\mathcal{H}_t(\\mathcal{A})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-16T10:17:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E35",
      "16T10",
      "18E10"
    ],
    "keywords": [
      "derived hall algebra",
      "duke math",
      "finitary hereditary abelian category",
      "associativity",
      "converse statement holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}